Collapse Theories (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeCollapse TheoriesFirst published Thu Mar 7, 2002; substantive revision Thu Jun 28, 2007Quantum mechanics, with its revolutionary implications, has posedinnumerable problems to philosophers of science. In particular, it hassuggested reconsidering basic concepts such as the existence of a worldthat is, at least to some extent, independent of the observer, thepossibility of getting reliable and objective knowledge about it, andthe possibility of taking (under appropriate circumstances) certainproperties to be objectively possessed by physical systems. It has alsoraised many others questions which are well known to those involved inthe debate on the interpretation of this pillar of modern science. Onecan argue that most of the problems are not only due to the intrinsicrevolutionary nature of the phenomena which have led to the developmentof the theory. They are also related to the fact that, in its standardformulation and interpretation, quantum mechanics is a theory which isexcellent (in fact it has met with a success unprecedented in thehistory of science) in telling us everything about what weobserve, but it meets with serious difficulties in telling uswhat is. We are making here specific reference to the centralproblem of the theory, usually referred to as the measurementproblem, or, with a more appropriate term, as themacro-objectification problem. It is just one of the manyattempts to overcome the difficulties posed by this problem that hasled to the development of Collapse Theories, i.e., to theDynamical Reduction Program (DRP). As we shall see, thisapproach consists in accepting that the dynamical equation of thestandard theory should be modified by the addition of stochastic andnonlinear terms. The nice fact is that the resulting theory is capable,on the basis of a unique dynamics which is assumed to govern allnatural processes, to account at the same time for all well-establishedfacts about microscopic systems as described by the standard theory aswell as for the so-called postulate of wave packet reduction (WPR). Asis well known, such a postulate is assumed in the standard scheme justin order to guarantee that measurements have outcomes but, aswe shall discuss below, it meets with insurmountable difficulties ifone takes the measurement itself to be a process governed by the linearlaws of the theory. Finally, the collapse theories account in acompletely satisfactory way for the classical behavior of macroscopicsystems. Two specifications are necessary in order to make clear from thebeginning what are the limitations and the merits of the program. Theonly satisfactory explicit models of this type (which are essentiallyvariations and refinements of the one, usually referred to as the GRWtheory, proposed in refs. [Ghirardi, Rimini and Weber, 1985, 1986])are phenomenological attempts to solve a foundational problem. Atpresent, they involve phenomenological parameters which, if the theoryis taken seriously, acquire the status of new constants ofnature. Moreover, the problem of building satisfactory relativisticgeneralizations of these models has encountered serious mathematicaldifficulties due to the appearance of untractable divergences. Onlyvery recently, some important steps we will discuss in what followshave led to the first satisfactory formulations of a genuinelyrelativistically invariant version of a dynamical reduction model,which, however, covers only the case of noninteracting particles.More important, the debate raised by these attempts and by claims thatthe desired generalization is impossible to achieve have elucidatedsome crucial points and have made clear that there is no reason ofprinciple preventing to reach this goal.In spite of their phenomenological character, we think that CollapseTheories have a remarkable relevance, since they have made clear thatthere are new ways to overcome the difficulties of the formalism, toclose the circle in the precise sense defined by AbnerShimony [Shimony, 1989], ways which until a few years ago wereconsidered impracticable, and which, on the contrary, have been shownto be perfectly viable. Moreover, they have allowed a clearidentification of the formal features which should characterize anyunified theory of micro and macro processes. Last but not least,Collapse theories qualify themselves as rival theories of quantummechanics and one can easily identify some of their physicalimplications which, in principle, would allow crucial testsdiscriminating between the two. This possibility, for the moment,seems to require experiments which go beyond the present tecnologicalpossibilities. However two aspects of the problem have to be takeninto account: due to the remarkable improvements in dealing withmesoscopic systems a crucial test of GRW might become feasible, andthe model suggests the kind of physical processes in which a violationof the linear nature of the formalism might occur. Accordingly, eventhough the experimental investigations might very well turn out not toconfirm the proposed new dynamical features of natural processes, theymight lead, in the end, to extremely relevant discoveries.1. General Considerations2. The Formalism: A Concise Sketch3. The Macro-Objectification Problem4. The Birth of Collapse Theories5. The Original Collapse Model6. The Continuous Spontaneous Localization Model (CSL)7. A Simplified Version of CSL8. Some remarks about Collapse Theories9. Relativistic Dynamical Reduction Models10. Collapse Theories and Definite Perceptions11. The Interpretation of the Theory and its Primitive Ontologies 12. The Problem of the Tails of the Wave Function 13. The Status of Collapse Models and Recent Positions about themSummaryBibliographyOther Internet ResourcesRelated Entries1. General ConsiderationsAs stated already, a very natural question which all scientists who areconcerned about the meaning and the value of science have to face, iswhether one can develop a coherent worldview that can accommodate ourknowledge concerning natural phenomena as it is embodied in our besttheories. Such a program meets serious difficulties with quantummechanics, essentially because of two formal aspects of the theorywhich are common to all of its versions, from the originalnonrelativistic formulations of the 1920s, to the quantum fieldtheories of recent years: the linear nature of the state space and ofthe evolution equation, i.e., the validity of the superpositionprinciple and the related phenomenon of entanglement, which, inSchrödinger's words: is not one but the characteristic trait of quantummechanics, the one that enforces its entire departure from classicallines of thought [Schrödinger, 1935, p. 807].These two formal features have embarrassing consequences, since theyimply objective chance in natural processes, i.e., the nonepistemicnature of quantum probabilities;objective indefiniteness of physical properties both at the microand macro level;objective entanglement between spatially separated andnon-interacting constituents of a composite system, entailing a sort ofholism and a precise kind of nonlocality.For the sake of generality, we shall first of all present a veryconcise sketch of ‘the rules of the game’.2. The Formalism: A Concise SketchLet us recall the axiomatic structure of quantum theory: 1.States of physical systems are associated withnormalized vectors in a Hilbert space, a complex, infinite-dimensional, complete and separablelinear vector space equipped with a scalar product. Linearity impliesthat the superposition principle holds: if |f> is a stateand |g> is a state, then (for a and barbitrary complex numbers) also |K> = a|f> +b|g>is a state. Moreover, the state evolution is linear, i.e., it preservessuperpositions: if |f,t> and|g,t> are the states obtained by evolving thestates |f,0> and |g,0>, respectively, from theinitial time t=0 to the time t, thena|f,t> +b|g,t> is the state obtained by theevolution of a|f,0> + b|g,0>.Finally, the completeness assumption is made, i.e., that the knowledgeof its statevector represents, in principle, the most accurateinformation one can have about the state of an individual physicalsystem.2.The observable quantities are represented byself-adjoint operators B on the Hilbert space. The associatedeigenvalue equations B|bk> =bk|bk> andthe corresponding eigenmanifolds (the linear manifolds spanned by theeigenvectors associated to a given eigenvalue, also called eigenspaces)play a basic role for the predictive content of the theory. Infact:i.The eigenvalues bk of anoperator B represent the only possible outcomes in a measurement of thecorresponding observable.ii.The norm (i.e. the length) of the projection of thenormalized vector (i.e. of length 1) describing the state of the systemonto the eigenmanifold associated to a given eigenvalue gives theprobability of obtaining the corresponding eigenvalue as the outcome ofthe measurement. In particular, it is useful to recall that when one isinterested in the probability of finding a particle at a given place,one has to resort to the so-called configuration space representationof the statevector. In such a case the statevector becomes asquare-integrable function of the position variables of the particlesof the system, whose modulus squared yields the probability density forthe outcomes of position measurements.We stress that, according to the above scheme, quantum mechanicsmakes only conditional probabilistic predictions (conditional on themeasurement being actually performed) for the outcomes of prospective(and in general incompatible) measurement processes. Only if a statebelongs already before the act of measurement to an eigenmanifold ofthe observable which is going to be measured, can one predict theoutcome with certainty. In all other cases — if the completenessassumption is made — one has objective nonepistemic probabilitiesfor different outcomes.The orthodox position gives a very simple answer to the question:what determines the outcome when different outcomes are possible?Nothing — the theory is complete and, as a consequence, it isillegitimate to raise any question about possessed properties referringto observables for which different outcomes have non-vanishingprobabilities of being obtained. Correspondingly, the referent of thetheory are the results of measurement procedures. These are to bedescribed in classical terms and involve in general mutually exclusivephysical conditions.As regards the legitimacy of attributing properties to physicalsystems, one could say that quantum mechanics warns us againstrequiring too many properties to be actually possessed by physicalsystems. However — with Einstein — one can adopt as asufficient condition for the existence of an objective individualproperty that one be able (without in any way disturbing the system)to predict with certainty the outcome of a measurement. This impliesthat, whenever the overall statevector factorizes into the product ofa state of the Hilbert space of the physical system S and ofthe rest of the world, S does possess some properties(actually a complete set of properties, i.e., those associated to amaximal set of commuting observables).Before concluding this section we must add some comments about themeasurement process. Quantum theory was created to deal withmicroscopic phenomena. In order to obtain information about them onemust be able to establish strict correlations between the states of themicroscopic systems and the states of objects we can perceive. Withinthe formalism, this is described by considering appropriate micro-macrointeractions. The fact that when the measurement is completed one canmake statements about the outcome is accounted for by the alreadymentioned WPR postulate [Dirac, 1948]: a measurement always causesa system to jump in an eigenstate of the observed quantity.Correspondingly, also the statevector of the apparatus‘jumps’ into the manifold associated to the recordedoutcome.3. The Macro-Objectification ProblemIn this Section we shall clarify why the formalism we have justpresented gives rise to the measurement or macro-objectificationproblem. To this purpose we shall, first of all, discuss the standardoversimplified argument based on the so-called von Neumann idealmeasurement scheme. Then we shall discuss more recent results [Bassiand Ghirardi, 2000], which relinquish von Neumann's assumptions.Let us begin by recalling the basic points of the standardargument:Suppose that a microsystem S, just before themeasurement of an observable B, is in the eigenstate|bj> of the corresponding operator. Theapparatus (a macrosystem) used to gain information about B isinitially assumed to be in a precise macroscopic state, its readystate, corresponding to a definite macro property — e.g., itspointer points at 0 on a scale. Since the apparatus A is madeof elementary particles, atoms and so on, it must be described byquantum mechanics, which will associate to it the state vector|A0>. One then assumes that there is anappropriate system-apparatus interaction lasting for a finite time,such that when the initial apparatus state is triggered by the state|bj> it ends up in a finalconfiguration |Aj>, which ismacroscopically distinguishable from the initial one and from the otherconfigurations |Ak> in which it wouldend up if triggered by a different eigenstate|bk>. Moreover, one assumes that thesystem is left in its initial state. In brief, one assumes that one candispose things in such a way that the system-apparatus interaction canbe described as: (1)(initial state):|bk>|A0> (final state):|bk>|Ak>Equation (1) and the hypothesis that the superposition principlegoverns all natural processes tell us that, if the initial state of themicrosystem is a linear superposition of different eigenstates (forsimplicity we will consider only two of them), one has:(2)(initial state):(a|bk> +b|bj>)|A0> (final state):(a|bk>|Ak>+b|bj>|Aj>).Some remarks about this are in order:The scheme is highly idealized, both because it takes for grantedthat one can prepare the apparatus in a precise state, which isimpossible since we cannot have control over all its degrees offreedom, and because it assumes that the apparatus registers theoutcome without altering the state of the measured system. However, aswe shall discuss below, these assumptions are by no means essential toderive the embarrassing conclusion we have to face, i.e., that thefinal state is a linear superposition of two states corresponding totwo macroscopically different states of the apparatus. Since we knowthat the + representing linear superpositions cannot be replaced by thelogical alternative either … or, the measurementproblem arises: what meaning can one attach to a state of affairs inwhich two macroscopically and perceptively different states occursimultaneously?As already mentioned, the standard solution to this problem isgiven by the WPR postulate: in a measurement process reduction occurs:the final state is not the one appearing at the right hand side. ofEq.(2) but, since macro-objectification takes place, it is (3) either|bk>|Ak>or|bj>|Aj>with probabilities |a|2 and|b|2, respectively.Nowadays, there is a general consensus that this solution isabsolutely unacceptable for two basic reasons:It corresponds to assuming that the linear nature of the theory isbroken at a certain level. Thus, quantum theory is unable to explainhow it can happen that the apparata behave as required by the WPRpostulate (which is one of the axioms of the theory).Even if one were to accept that quantum mechanics has a limitedfield of applicability, so that it does not account for all naturalprocesses and, in particular, it breaks down at the macrolevel, it isclear that the theory does not contain any precise criterion foridentifying the borderline between micro and macro, linear andnonlinear, deterministic and stochastic, reversible and irreversible.To use J.S. Bell's words, there is nothing in the theory fixing such aborderline and the split between the two above types ofprocesses is fundamentally shifty. As a matter of fact, if onelooks at the historical debate on this problem, one can easily see thatit is precisely by continuously resorting to this ambiguity about thesplit that adherents of the Copenhagen orthodoxy or easysolvers [Bell, 1990] of the measurement problem have rejected thecriticism of the heretics [Gottfried, 2000]. For instance,Bohr succeeded in rejecting Einstein's criticisms at the SolvayConferences by stressing that some macroscopic parts of the apparatushad to be treated fully quantum mechanically; von Neumann and Wignerdisplaced the split by locating it between the physical and theconscious (but what is a conscious being?), and so on. Also otherproposed solutions to the problem, notably certain versions ofmany-worlds interpretations, suffer from analogous ambiguities.It is not our task to review here the various attempts to solve theabove difficulties. One can find many exhaustive treatments of thisproblem in the literature. On the contrary, we would like to discusshow the macro-objectification problem is indeed a consequence of verygeneral, in fact unavoidable, assumptions on the nature ofmeasurements, and not specifically of the assumptions of von Neumann'smodel. This was established in a series of theorems of increasinggenerality, notably the ones by Fine [1970], d'Espagnat [1971], Shimony [1974], Brown[1986] and Busch and Shimony [1996]. Possibly the most general anddirect proof is given by Bassi and Ghirardi [2000], whose results webriefly summarize. The assumptions of the theorem are:(i) that a microsystem can be prepared in two differenteigenstates of an observable (such as, e.g., the spin component alongthe z-axis) and in a superposition of two such states; (ii) that one has a sufficiently reliable way of‘measuring’ such an observable, meaning that when themeasurement is triggered by each of the two above eigenstates, theprocess leads in the vast majority of cases to macroscopically andperceptually different situations of the universe. This requirementallows for cases in which the experimenter does not have perfectcontrol of the apparatus, the apparatus is entangled with the rest ofthe universe, the apparatus makes mistakes, or the measured system isaltered or even destroyed in the measurement process;(iii) that all natural processes obey the linear laws of thetheory.From these very general assumptions one can show that, repeating themeasurement on systems prepared in the superposition of the two giveneigenstates, in the great majority of cases one ends up in asuperposition of macroscopically and perceptually different situationsof the whole universe. If one wishes to have an acceptable finalsituation, one mirroring the fact that we have definite perceptions,one is arguably compelled to break the linearity of the theory at anappropriate stage.4. The Birth of Collapse TheoriesThe debate on the macro-objectification problem continued for manyyears after the early days of quantum mechanics. In the early 1950s animportant step was taken by D. Bohm who presented [Bohm, 1952] amathematically precise deterministic completion of quantum mechanics(see the entry on Bohmian Mechanics). In the area of Collapse Theories,one should mention the contribution by Bohm and Bub [1966], which wasbased on the interaction of the statevector with Wiener — Siegelhidden variables. But let us come to Collapse Theories in the sensecurrently attached to this expression. Various investigations during the 1970s can be considered aspreliminary steps for the subsequent developments. In the years1970-1973 L. Fonda, A. Rimini, T. Weber and myself were seriouslyconcerned with quantum decay processes and in particular with thepossibility of deriving, within a quantum context, the exponentialdecay law [Fonda, Ghirardi, Rimini and Weber; 1973, Fonda etal., 1978]. Some features of this approach are extremely relevantfor the DRP. Let us list them:One deals with individual physical systems;The statevector is supposed to undergo random processes at randomtimes, inducing sudden changes driving it either within the linearmanifold of the unstable state or within the one of the decayproducts;To make the treatment quite general (the apparatus does not knowwhich kind of unstable system it is testing) one is led to identify therandom processes with localization processes of the relativecoordinates of the decay fragments. Such an assumption, combined withthe peculiar resonant dynamics characterizing an unstable system,yields, completely in general, the desired result. The ‘relativeposition basis’ is the preferred basis of this theory;We have applied analogous ideas to measurement processes [Fonda,Ghirardi and Rimini, 1973];The final equation for the evolution at the ensemble level is ofthe quantum dynamical semigroup type and has a structure extremelysimilar to the final one of the GRW theory.Obviously, in these papers the reduction processes which are involvedwere not assumed to be ‘spontaneous and fundamental’natural processes, but due to system-environmentinteractions. Accordingly, these attempts did not represent originalproposals for solving the macro-objectification problem but they havepaved the way for the elaboration of the GRW theory. Almost in the same years, P. Pearle [Pearle, 1976,1979], andsubsequently N. Gisin [Gisin, 1984] and others, had entertained theidea of accounting for the reduction process in terms of a stochasticdifferential equation. These authors were really looking for a newdynamical equation and for a solution to the macro-objectificationproblem. Unfortunately, they have not been able to give any precisesuggestion about how to identify the states to which the dynamicalequation should lead. Indeed, these states were assumed to depend onthe particular measurement process one was considering. Without aclear indication on this point there was no way to identify amechanism whose effect could be negligible for microsystems butextremely relevant for the macroscopic ones. N. Gisin gavesubsequently an interesting (though not uncontroversial) argument[Gisin, 1989] that nonlinear modifications of the standard equationwithout stochasticity are unacceptable since they imply thepossibility of sending superluminal signals. Soon afterwards,R. Grassi and myself [Ghirardi and Grassi, 1991] showed thatstochastic modifications without nonlinearity can at most induceensemble and not individual reductions, i.e., they do not guaranteethat the state vector of each individual physical system is driven ina manifold corresponding to definite properties.5. The Original Collapse ModelAs already mentioned, the Collapse Theory [Ghirardi, Rimini and Weber,1986] we are going to describe amounts to accepting a modification ofthe standard evolution law of the theory such that microprocesses andmacroprocesses are governed by a unique dynamics. Such a dynamics mustimply that the micro-macro interaction in a measurement process leadsto WPR. Bearing this in mind, recall that the characteristic featuredistinguishing quantum evolution from WPR is that, whileSchrödinger's equation is linear and deterministic (at the wavefunction level), WPR is nonlinear and stochastic. It is then naturalto consider, as was suggested for the first time in the above quotedpapers by P. Pearle, the possibility of nonlinear and stochasticmodifications of the standard Schrödinger dynamics. However, theinitial attempts to implement this idea were unsatisfactory forvarious reasons. The first, which we have already discussed, concernsthe choice of the preferred basis: if one wants to have a universalmechanism leading to reductions, to which linear manifolds should thereduction mechanism drive the statevector? Or, equivalently, which ofthe (generally) incompatible ‘potentialities’ of thestandard theory should we choose to make actual? The second, referredto as the trigger problem by Pearle [Pearle, 1989], is the problem ofhow the reduction mechanism can become more and more effective ingoing from the micro to the macro domain. The solution to this problemconstitutes the central feature of the Collapse Theories of the GRWtype. To discuss these points, let us briefly review the firstconsistent Collapse model [Ghirardi, Rimini and Weber, 1985] to appearin the literature.Within such a model, originally referred to as QMSL (QuantumMechanics with Spontaneous Localizations), the problem of the choice ofthe preferred basis is solved by remarking that the most embarrassingsuperpositions, at the macroscopic level, are those involving differentspatial locations of macroscopic objects. Actually, as Einstein hasstressed [Born, 1971, p. 223], this is a crucial point which has to befaced by anybody aiming to take a macro-objective position aboutnatural phenomena: ‘A macro-body must always have a quasi-sharplydefined position in the objective description of reality’.Accordingly, QMSL considers the possibility of spontaneous processes,which are assumed to occur instantaneously and at the microscopiclevel, which tend to suppress the linear superpositions of differentlylocalized states. The required trigger mechanism must then followconsistently.The key assumption of QMSL is the following: each elementaryconstituent of any physical system is subjected, at random times, torandom and spontaneous localization processes (which we will callhittings) around appropriate positions. To have a precise mathematicalmodel one has to be very specific about the above assumptions; inparticular one has to make explicit HOW the process works, i.e. whichmodifications of the wave function are induced by the localizations,WHERE it occurs, i.e. what determines the occurrence of a localizationat a certain position rather than at another one, and finally WHEN,i.e. at what times, it occurs. The answers to these questions are asfollows.Let us consider a system of N distinguishable particles andlet us denote by F(q1,q2, … ,qN) the coordinaterepresentation (wave function) of the state vector (we disregard spinvariables since hittings are assumed not to act on them).(a) The answer to the question HOW is then: if a hittingoccurs for the i-th particle at pointx, the wave function is instantaneouslymultiplied by a Gaussian function (appropriately normalized) G(qi,x) = K exp[−{1/(2d2)}(qi−x)2],where d represents the localization accuracy. Let us denote asLi(q1,q2, … ,qN ;x) =F(q1,q2, … ,qN)G(qi,x)the wave function immediately after the localization, as yetunnormalized.(b) As concerns the specification of WHERE the localization occurs,it is assumed that the probability densityP(x) of its taking place at thepoint x is given by the norm of the stateLi (the length, or to be more precise, theintegral of the modulus squared of the functionLi over the 3N-dimensionalspace). This implies that hittings occur with higher probability atthose places where, in the standard quantum description, there is ahigher probability of finding the particle. Note that the aboveprescription introduces nonlinear and stochastic elements in thedynamics. The constant K appearing in the expression ofG(qi,x) is chosen in such a way that the integral ofP(x) over the whole space equals1.(c) Finally, the question WHEN is answered by assuming that thehittings occur at randomly distributed times, according to a Poissondistribution, with mean frequency f.It is straightforward to convince oneself that the hitting processleads, when it occurs, to the suppression of the linear superpositionsof states in which the same particle is well localized at differentpositions separated by a distance greater than d. As a simpleexample we can consider a single particle whose wavefunction isdifferent from zero only in two small and far apart regions hand t. Suppose that a localization occurs around h;the state after the hitting is then appreciably different from zeroonly in a region around h itself. A completely analogousargument holds for the case in which the hitting takes place aroundt. As concerns points which are far from both h andt, one easily sees that the probability density for suchhittings , according to the multiplication rule determiningLi, turns out to be practically zero, andmoreover, that if such a hitting were to occur, after the wave functionis normalized, the wave function of the system would remain almostunchanged.We can now discuss the most important feature of the theory, i.e.,the Trigger Mechanism. To understand the way in which the spontaneouslocalization mechanism is enhanced by increasing the number ofparticles which are in far apart spatial regions (as compared tod), one can consider, for simplicity, the superposition|S>, with equal weights, of two macroscopic pointer states|H> and |T>, corresponding to two differentpointer positions H and T, respectively. Taking intoaccount that the pointer is ‘almost rigid’ and contains amacroscopic number N of microscopic constituents, the statecan be written, in obvious notation, as:(4) |S> = [|1 near h1>… |N near hN> + |1 neart1> … |N neartN>],where hi is near H, andti is near T. The statesappearing in first term on the right-hand side of Eq. (4) havecoordinate representations which are different from zero only whentheir arguments (1,…,N) are all near H, whilethose of the second term are different from zero only when they are allnear T. It is now evident that if any of the particles (say,the i-th particle) undergoes a hitting process, e.g. near thepoint hi, the multiplication prescriptionleads practically to the suppression of the second term in (4). Thusany spontaneous localization of any of the constituents amounts to alocalization of the pointer. The hitting frequency is thereforeeffectively amplified proportionally to the number of constituents.Notice that, for simplicity, the argument makes reference to an almostrigid body, i.e. to one for which all particles are around Hin one of the states of the superposition and around T in theother. It should however be obvious that what really matters inamplifying the reductions is the number of particles which are indifferent positions in the two states appearing in the superpositionitself.Under these premises we can now proceed to choose the parametersd and f of the theory, i.e., the localizationaccuracy and the mean localization frequency. The argument just givenallows one to understand how one can choose the parameters in such away that the quantum predictions for microscopic systems remain fullyvalid while the embarrassing macroscopic superpositions inmeasurement-like situations are suppressed in very short times.Accordingly, as a consequence of the unified dynamics governing allphysical processes, individual macroscopic objects acquire definitemacroscopic properties. The choice suggested in the GRW-model is:(5)f = 10-16 s-1 d = 10-5 cmIt follows that a microscopic system undergoes a localization, onaverage, every hundred million years, while a macroscopic one undergoesa localization every 10-7 seconds. With reference to thechallenging version of the macro-objectification problem presented bySchrödinger with the famous example of his cat, J.S. Bell comments[Bell, 1987, p.44]: [within QMSL] the cat is not both dead andalive for more than a split second . Besides the extremely lowfrequency of the hittings for microscopic systems, also the fact thatthe localization width is large compared to the dimensions of atoms (sothat even when a localization occurs it does very little violence tothe internal economy of an atom) plays an important role inguaranteeing that no violation of well-tested quantum mechanicalpredictions is implied by the modified dynamics.Some remarks are appropriate. First of all, QMSL, being preciselyformulated, allows to locate precisely the ‘split’ betweenmicro and macro, reversible and irreversible, quantum and classical.The transition between the two types of ‘regimes’ isgoverned by the number of particles which are well localized atpositions further apart than 10-5 cm in the two states whosecoherence is going to be dynamically suppressed. Second, the model is,in principle, testable against quantum mechanics. As a matter of fact,an essential part of the program consists in proving that itspredictions do not contradict any established fact about microsystemsand macrosystems.6. The Continuous Spontaneous Localization Model (CSL)The model just presented (QMSL) has a serious drawback: it does notallow to deal with systems containing identical constituents becauseit does not respect the symmetry or antisymmetry requirements for suchparticles. A quite natural idea to overcome this difficulty would bethat of relating the hitting process not to the individual particlesbut to the particle number density averaged over an appropriatevolume. This can be done by introducing a new phenomenologicalparameter in the theory which however can be eliminated by anappropriate limiting procedure (see below).Another way to overcome this problem derives from injecting thephysically appropriate principles of the GRW model within the original approachof P. Pearle. This line of thought has led to a quite elegantformulation of a dynamical reduction model, usually referred to as CSL[Pearle, 1989; Ghirardi, Pearle and Rimini, 1990] in which thediscontinuous jumps which characterize QMSL are replaced by acontinuous stochastic evolution in the Hilbert space (a sort ofBrownian motion of the statevector).We will not enter into the rather technical details of thisinteresting development of the original GRW proposal, since the basicideas and physical implications are precisely the same as those of theoriginal formulation. Actually, one could argue that the above idea oftackling the problem of identical particles by considering the averageparticle number within an appropriate volume is correct. In fact it hasbeen proved [Ghirardi, Pearle and Rimini, 1990] that for any CSLdynamics there is a hitting dynamics which, from a physical point ofview, is ‘as close to it as one wants’. Instead of enteringinto the details of the CSL formalism, it is useful, for the discussionbelow, to analyze a simplified version of it.7. A Simplified Version of CSLWith the aim of understanding the physical implications of the CSLmodel, such as the rate of suppression of coherence, we make now somesimplifying assumptions. First, we assume that we are dealing with onlyone kind of particles (e.g., the nucleons), secondly, we disregard thestandard Schrödinger term in the evolution and, finally, we dividethe whole space in cells of volume d3. We denote by|n1, n2, … > a statein which there are ni particles in cell i,and we consider a superposition of two states |n1,n2, … > and |m1,m2, … > which differ in the occupationnumbers of the various cells of the universe. With these assumptions itis quite easy to prove that the rate of suppression of the coherencebetween the two states (so that the final state is one of the two andnot their superposition) is governed by the quantity: (6) exp{−f [(n1 −m1)2 + (n2 −m2)2 +…]t},the sum being extended to all cells in the universe. Apart fromdifferences relating to the identity of the constituents, the overallphysics is quite similar to that implied by QMSL. Obviously, there areinteresting physical implications which deserve to be discussed. Adetailed analysis has been presented in [Ghirardi and Rimini,1990]. As shown there and as follows from estimates about possibleeffects for superconducting devices [Rae, 1990; Gallis and Fleming,1990; Rimini, 1995], and for the excitation of atoms [Squires, 1991],it turns out not to be possible, with present technology, to performclear-cut experiments allowing to discriminate the model from standardquantum mechanics [Benatti et al., 1995]. Very recently, aninteresting proposal of testing the superposition principle byresorting to an experimental set-up involving a (mesoscopic) mirrorhas been advanced [Marshall et al., 2003]. This stimulatingproposal has led a group of scientists directly interested in CollapseTheories [Bassi et al., 2005] to check whether the proposedexperiment might be a crucial one for dynamical reduction modelsversus quantum mechanics. The rigorous conclusion has been that thisis not the case: in the devised situation the GRW and CSL theorieshave implications which agree with those of the standard theory.There is however an interesting aspect which might be relevant to theidea of relating the suppression of coherence to gravitational effects.Given Eq. (6), notice that the worst case scenario (from the point ofview of the time necessary to suppress coherence) is the superpositionof two states for which the occupation numbers of the individual cellsdiffer only by one unit. Indeed, in this case the amplifying effect oftaking the square of the differences disappears. Let us then raise thequestion: how many nucleons (at worst) should occupy different cells,in order for the given superposition to be dynamically suppressedwithin the time which characterizes human perceptual processes? Sincesuch a time is of the order of 10-2 sec and f =10-16 sec-1, the number of displaced nucleonsmust be of the order of 1018, which corresponds, to aremarkable accuracy, to a Planck mass. This figure seems to point inthe same direction as attempts such as Penrose's attempts to relatereduction mechanisms to quantum gravitational effects [Penrose,1989].8. Some remarks about Collapse TheoriesA. Pais famously recalls in his biography of Einstein: We often discussed his notions on objective reality. Irecall that during one walk Einstein suddenly stopped, turned to me andasked whether I really believed that the moon exists only when I lookat it [Pais, 1982, p. 5].In the context of Einstein's remarks in Albert Einstein,Philosopher-Scientist [Schilpp, 1949], we can regard thisreference to the moon as an extreme example of ‘a fact thatbelongs entirely within the sphere of macroscopic concepts’, asis also a mark on a strip of paper that is used to register the outcomeof a decay experiment, so that as a consequence, there is hardly likely to be anyone whowould be inclined to consider seriously […] that the existenceof the location is essentially dependent upon the carrying out of anobservation made on the registration strip. For, in the macroscopicsphere it simply is considered certain that one must adhere to theprogram of a realistic description in space and time; whereas in thesphere of microscopic situations one is more readily inclined to giveup, or at least to modify, this program [p. 671].However, the ‘macroscopic’ and the‘microscopic’ are so inter-related that it appearsimpracticable to give up this program in the ‘microscopic’alone [p. 674].One might speculate that Einstein would not have taken the DRPseriously, given that it is a fundamentally indeterministicprogram. On the other hand, the DRP allows precisely for this middleground, between giving up a ‘classical description in space andtime’ altogether (the moon is not there when nobody looks), andrequiring that it be applicable also at the microscopic level (aswithin some kind of ‘hidden variables’ theory). It wouldseem that the pursuit of ‘realism’ for Einstein was more aprogram that had been very successful rather than an a prioricommitment, and that in principle he would have accepted attemptsrequiring a radical change in our classical conceptions concerningmicrosystems, provided they would nevertheless allow to take amacrorealist position matching our definite perceptions at thisscale.In the DRP, we can say of an electron in an EPR-Bohm situation that‘when nobody looks’, it has no definite spin in anydirection , and in particular that when it is in a superposition oftwo states localised far away from each other, it cannot be thought tobe at a definite place (see, however, the remarks in Section 11). Inthe macrorealm, however, objects do have definite positions and aregenerally describable in classical terms. That is, in spite of thefact that the DRP program is not adding ‘hidden variables’to the theory, it implies that the moon is definitely there even if nosentient being has ever looked at it. In the words of J. S. Bell, theDRPallows electrons (in general microsystems) to enjoy thecloudiness of waves, while allowing tables and chairs, and ourselves,and black marks on photographs, to be rather definitely in one placerather than another, and to be described in classical terms [Bell,1986, p. 364].Such a program, as we have seen, is implemented by assuming only theexistence of wave functions, and by proposing a unified dynamics thatgoverns both microscopic processes and ‘measurements’. Asregards the latter, no vague definitions are needed. The new dynamicalequations govern the unfolding of any physical process, and themacroscopic ambiguities that would arise from the linear evolution aretheoretically possible, but only of momentary duration, of nopractical importance and no source of embarrassment.We have not yet analyzed the implications about locality, but since inthe DRP program no hidden variables are introduced, the situation canbe no worse than in ordinary quantum mechanics: ‘by addingmathematical precision to the jumps in the wave function, it simplymakes precise the action at a distance of ordinary quantummechanics’ [Bell, 1987, p. 46]. Indeed, a detailedinvestigation of the locality properties of the theory becomespossible, and the analysis carried on so far proves that at least inthe non-relativistic version a program of dynamical reduction can beconsistently developed. Moreover, as it will become clear when we willdiscuss the interpretation of the theory in terms of mass density, theQMSL and CSL theories lead in a natural way to account for a behaviourof macroscopic objects corresponding to our definite perceptions aboutthem, the main objective of Einstein's requirements.The achievements of the DRP which are relevant for the debate aboutthe foundations of quantum mechanics can also be concisely summarizedin the words of H.P. Stapp:The collapse mechanisms so far proposed could, on the onehand, be viewed as ad hoc mutilations designed to force ontology tokneel to prejudice. On the other hand, these proposals show that onecan certainly erect a coherent quantum ontology that generally conformsto ordinary ideas at the macroscopic level [Stapp, 1989, p.157].9. Relativistic Dynamical Reduction ModelsAs soon as our proposal appeared and attracted the attention ofJ.S. Bell it also stimulated him to look at it from the point of viewof relativity theory. As he stated subsequently [Bell, 1989]:When I saw this theory first, I thought that I could blowit out of the water, by showing that it was grossly inviolation of Lorentz invariance. That's connected with the problem of‘quantum entanglement’, the EPR paradox.Actually, he had already investigated this point by studying theeffect on the theory of a transformation mimicking a nonrelativisticapproximation of a Lorentz transformation and he arrived [Bell, 1987]at a surprising conclusion:… the model is as Lorentz invariant as it could bein its nonrelativistic version. It takes away the ground of my fearthat any exact formulation of quantum mechanics must conflict withfundamental Lorentz invariance.What Bell had actually proved in a rather complicated way by resortingto a two-times formulation of the Schrödinger equation is thatthe model violates locality by violating outcome independence and not,as deterministic hidden variable theories do, parameter independence.Indeed, with reference to this point we recall that, as is well known,[Suppes and Zanotti, 1976; van Fraassen, 1982; Jarrett, 1984; Shimony,1983; see also the entry on Bell's Theorem], Bell's localityassumption is equivalent to the conjunction of two other assumptions,viz., in Shimony's terminology, parameter independence and outcomeindependence. In view of the experimental violation of Bell'sinequality, one has to give up either or both of theseassumptions. The above splitting of the locality requirement into twologically independent conditions is particularly useful in discussingthe different status of CSL and deterministic hidden variable theorieswith respect to relativistic requirements. Actually, as proved byJarrett himself, when parameter independence is violated, if one hadaccess to the variables which specify completely the state ofindividual physical systems, one could send faster-than-light signalsfrom one wing of the apparatus to the other. Moreover, in refs.[Ghirardi and Grassi, 1994, 1996] it has been proved that it isimpossible to build a genuinely relativistically invarianttheory which, in its nonrelativistic limit, exhibits parameterdependence. Here we use the term genuinely invariant todenote a theory for which there is no (hidden) preferred referenceframe. On the other hand, if locality is violated only by theoccurrence of outcome dependence then faster-than-light signalingcannot be achieved [Eberhard, 1978; Ghirardi, Rimini and Weber, 1980;Ghirardi, Grassi, Rimini and Weber, 1988]. Few years after the justmentioned proof by Bell, it has been shown in complete generality[Ghirardi, Grassi, Butterfield and Fleming, 1993; Butterfield etal., 1993] that the GRW and CSL theories, just as standardquantum mechanics, exhibit only outcome dependence. This is to someextent encouraging and shows that there are no reasons of principlemaking unviable the project of building a relativistically invariantDRM.Let us be more specific about this crucial problem. P. Pearle was thefirst to propose [Pearle, 1990] a relativistic generalization of CSLto a quantum field theory describing a fermion field coupled to ameson scalar field enriched with the introduction of stochastic andnonlinear terms. A quite detailed discussion of this proposal waspresented in [Ghirardi et al,1990a] where it was shown thatthe theory enjoys of all properties which are necessary in order tomeet the relativistic constraints. Pearle's approach requires theprecise formulation of the idea of stochastic Lorentz invariance. Theproposal can be summarized in the following terms. One considers a fermion field coupled to a meson field and putsforward the idea of inducing localizations for the fermions throughtheir coupling to the mesons and a stochastic dynamical reductionmechanism acting on the meson variables. In practice, one considersHeisenberg evolution equations for the coupled fields and aTomonaga-Schwinger CSL-type evolution equation with a skew-hermitiancoupling to a c-number stochastic potential for the state vector. Thisapproach has been systematically investigated in refs. [Ghirardi,Grassi and Pearle, 1990a, 1990b] to which we refer the reader for adetailed discussion. Here we limit ourselves to stressing that, undercertain approximations, one obtains in the non-relativistic limit aCSL-type equation inducing spatial localization. However, due to thewhite noise nature of the stochastic potential, novel renormalizationproblems arise: the increase per unit time and per unit volume of theenergy of the meson field is infinite due to the fact that infinitelymany mesons are created. For these reasons one cannot consider this asa satisfactory example of a relativistic reduction model.The years following the original attempts just mentioned saw afluorishing of researches aimed to get the desired result, most ofthem having been proposed by P. Pearle himself. However, no real stepsforward were made concerning the suppression of the untractabledivergencies as plainly recognized by Pearle himself [Pearle,2006]. As already mentioned, there are indications that theirappearence is due to the white character of the noise which accountsfor the stochastic nature of the evolution. For this reason P. Pearle[Pearle, 1999], L. Diosi [Diosi, 1990] and A. Bassi and myself [Bassiet al., 2002] reconsidered the problem from the beginning byinvestigating nonrelativistic theories with nonwhite Gaussiannoises. It is not yet clear whether this approach will lead to a realstep forward. It is interesting to remark that precisely in the same years similarattempts to get a relativistic generalization of the other existing‘ exact’ theory, i.e., Bohmian Mechanics, were going onand that they too have encountered serious difficulties. Importantsteps are represented by a paper [Dürr, 1999] resorting to apreferred spacetime slicing, by the investigations of Goldstein andTumulka [Goldstein 2003] and by other scientists [Berndl,1996]. However, we must recognize that no one of these attempts hasled to a fully satisfactory solution of the problem of having a theorywithout observers, like Bohmian mechanics, which is perfectlysatisfactory from the relativistic point of view.Let us come back to the relativistic DRP. With reference to it wemention an attempt by Dove and Squires [Dove,1996] based on discreterather than continuous stochastic processes and one by Dewdney andHorton [Dewdney,2001] formulated on a discrete space-time. All otherattempts towards a relativistic collapse model are based on acontinuous spontaneous localization description of the reductionmechasnism. Among them mention must be made of the investigations byPearle [Pearle, 1999b] and by Nicrosini and Rimini [Nicrosini,2003]. The first one, however, is not fully relativistic, and thesecond is based on a Tomonaga-Schwinger equation which turns out to benot integrable.Some important changes, in my opinion, have occurred quiterecently. Tumulka [Tumulka, 2006] succeeded in proposing arelativistic version of the GRW theory for N non-interactingdistinguishable particles, based on the consideration of a multi-timewavefunction whose evolution is governed by Dirac like equations andadopts as its Primitive Ontology (see the next section) the one whichattaches a primary role to the space and time points at whichspontaneous localizations occur, as originally suggested by Bell[Bell, 1987]. To my knowledge this represents the first proposal of arelativistic dynamical reduction mechanism which satisfies allrelativistic requirementes, even though it can deal only with systemsof noninteracting particles. His second step has been to present[Tumulka, 2006b] a quantum field theoretical model of dynamicalreduction. Presently he is trying to combine the two approaches to geta fully relativistic and realistic field theoretical scheme of aquantum mechanics without observers.I consider particularly interesting the conclusions which he drawsfrom his deep analysis concerning both Bohiam-like and GRW-likeapproaches to the relativistic macro-objectification process:A somewhat surprising feature of the present situation isthat we seem to arrive at the following alternative: Bohmian mechanicsshows that one can explain quantum mechanics, exactly and completely,if one is willing to pay with using a preferred slicing of spacetime;our model suggests that one should be able to avoid a preferredslicing of spacetime if one is willing to pay with a certain deviationfrom quantum mechanics, a conclusion that he has rephrased and reinforced in [Tumulka, 2006c]: Thus, with the presently available models we have thealternative: either the conventional understanding of relativity isnot right, or quantum mechanics is not exact. I believe that this position is perfectly consistent with the presentideas concerning the attempts to transform relativistic standardquantum mechanics into an ‘exact ’ theory in the sensewhich has been made precise by J. Bell. Since the only unified,mathematically precise and formally consistent formulations of thequantum description of natural processes are Bohmian mechanics andGRW-like theories, if one chooses the first alternative one has toaccept the existence of a preferred reference frame, while in thesecond case one is not led to such a drastic change of position withrespect to relativistic concepts but must accept that the ensuingtheory — even though only in a presently non-testable manner -disagrees with the predictions of quantum mechanics and acquires thestatus of a rival theory with respect to it.In spite of the above remarks, it has to be recognized that theefforts which have been spent on such a program have led to a betterunderstanding of some points and have thrown light on some importantconceptual issues. First, they have led to a completely general andrigorous formulation of the concept of stochastic invariance[Ghirardi, Grassi and Pearle, 1990a]. Second, they have prompted acritical reconsideration, based on the discussion of smearedobservables with compact support, of the problem of locality at theindividual level. This analysis has brought out the necessity ofreconsidering the criteria for the attribution of objective localproperties to physical systems. In specific situations, one cannotattribute any local property to a microsystem: any attempt to do sogives rise to ambiguities. However, in the case of macroscopicsystems, the impossibility of attributing to them local properties(or, equivalently, the ambiguity associated to such properties) lastsonly for time intervals of the order of those necessary for thedynamical reduction to take place. Moreover, no objective propertycorresponding to a local observable, even for microsystems, can emergeas a consequence of a measurement-like event occurring in a space-likeseparated region: such properties emerge only in the future light coneof the considered macroscopic event. Finally, recent investigations[Ghirardi and Grassi, 1994, 1996; Ghirardi, 1996, 2000] have shownthat the very formal structure of the theory is such that it does notallow, even conceptually, to establish cause-effect relations betweenspace-like events.Having listed some interesting results obtained along these lines, inconcluding this section it is necessary to stress once more that thequestion of whether a relativistic dynamical reduction program willfind a satisfactory formulation still remains ‘the bigproblem’ for this type of investigations.Recently, a paper by Conway and Kochen [Conway, 2006] which has raiseda lot of interest has been published, and we cannot avoid to spend fewwords on it to clarify possible misunderstandings. The first and mostimportant aim of the paper is the derivation of what the authors havecalled The Free Will Theorem , putting forward theprovocative idea that if human beings are free to make their choicesabout the measurements they will perform on one of a pair of far-awayentangled particles, then one must admit that also the elementaryparticles involved in the experiment have free will. One might makeseveral comments on this statement. For what concerns us here therelevant fact is that the authors claim that their theorem implies, asa byproduct, the impossibility of elaborating a relativisticallyinvariant dynamical reduction model. A lively debate has arosen; werefer the reader to the papers by Adler [Adler, 2006], Bassi andGhirardi [Bassi, 2007], Tumulka [Tumulka, 2007] in which it is provedthat the conclusion drawn by Conway and Kochen is not pertinent to theproblem. Very recently the above authors have replied [Conway etal., 2007] to all criticisms raised in the just mentioned papers,in a way which, in my opinon, is not correct. I am sure that, inprinciple, nothing forbids a relativistic generalization of the GRWtheory, and there are many elements which clearly point in thisdirection.10. Collapse Theories and Definite PerceptionsSome authors [Albert and Vaidman, 1989; Albert, 1990, 1992] haveraised an interesting objection concerning the emergence of definiteperceptions within Collapse Theories. The objection is based on thefact that one can easily imagine situations leading to definiteperceptions, that nevertheless do not involve the displacement of alarge number of particles up to the stage of the perception itself.These cases would then constitute actual measurement situations whichcannot be described by the GRW theory, contrary to what happens forthe idealized (according to the authors) situations considered in manypresentations of it, i.e. those involving the displacement of somesort of pointer. To be more specific, the above papers consider a‘measurement-like’ process whose output is the emission ofa burst of few photons triggered by the position in which a particlehits a screen. This can easily be devised by considering, e.g., aStern-Gerlach set-up in which the two paths followed by themicrosystem according to the value of its spin component hit afluorescent screen and excite a small number of atoms whichsubsequently decay, emitting a small number of photons. The argumentgoes as follows: if one triggers the apparatus with a superposition oftwo spin states, since only a few atoms are excited, since theexcitations involve displacements which are smaller than thecharacteristic localization distance of GRW, since GRW does not inducereductions on photon states and, finally, since the photon statesimmediately overlap, there is no way for the spontaneous localizationmechanism to become effective in suppressing the ensuing superpositionof the states ‘photons emerging from point A of thescreen’ and ‘photons emerging from point B of thescreen’. On the other hand, since the visual perceptionthreshold is quite low (about 6-7 photons), there is no doubt that thenaked eye of a human observer is sufficient to detect whether theluminous spot on the screen is at A or at B. Theconclusion follows: in the case under consideration no dynamicalreduction can take place and as a consequence no measurement is over,no outcome is definite, up to the moment in which a conscious observerperceives the spot.We have presented a detailed answer to this criticism [Aicardi etal., 1991]. The crucial points of our argument are the following:we perfectly agree that in the case considered the superpositionpersists for long times (actually the superposition must persist,since, the system under consideration being microscopic, one couldperform interference experiments which everybody would expect toconfirm quantum mechanics). However, to deal in the appropriate andcorrect way with such a criticism, one has to consider all the systemswhich enter into play (electron, screen, photons and brain) and theuniversal dynamics governing all relevant physical processes. A simpleestimate of the number of ions which are involved in the visualperception mechanism makes perfectly plausible that, in the process, asufficient number of particles are displaced by a sufficient spatialamount to satisfy the conditions under which, according to the GRWtheory, the suppression of the superposition of the two nervoussignals will take place within the time scale of perception.To avoid misunderstandings, this analysis by no means amounts toattributing a special role to the conscious observer or to perception.The observer's brain is the only system present in the set-up in whicha superposition of two states involving different locations of a largenumber of particles occurs. As such it is the only place where thereduction can and actually must take place according to the theory. Itis extremely important to stress that if in place of the eye of ahuman being one puts in front of the photon beams a spark chamber or adevice leading to the displacement of a macroscopic pointer, orproducing ink spots on a computer output, reduction will equally takeplace. In the given example, the human nervous system is simply aphysical system, a specific assembly of particles, which performs thesame function as one of these devices, if no other such deviceinteracts with the photons before the human observer does. It followsthat it is incorrect and seriously misleading to claim that the GRWtheory requires a conscious observer in order that measurements have adefinite outcome.A further remark may be appropriate. The above analysis could be takenby the reader as indicating a very naive and oversimplified attitudetowards the deep problem of the mind-brain correspondence. There isno claim and no presumption that GRW allows a physicalist explanationof conscious perception. It is only pointed out that, for what we knowabout the purely physical aspects of the process, one can state thatbefore the nervous pulses reach the higher visual cortex, theconditions guaranteeing the suppression of one of the two signals areverified. In brief, a consistent use of the dynamical reductionmechanism in the above situation accounts for the definiteness of theconscious perception, even in the extremely peculiar situation devisedby Albert and Vaidman.11. The Interpretation of the Theory and its Primitive Ontologies As stressed in the opening sentences of this contribution, the mostserious problem of standard quantum mechanics lies in its beingextremely successful in telling us about what we observe, butbeing basically silent on what is. This specific feature isclosely related to the probabilistic interpretation of thestatevector, combined with the completeness assumption of thetheory. Notice that what is under discussion is the probabilisticinterpretation, not the probabilistic character, of the theory. Alsocollapse theories have a fundamentally stochastic character, but, dueto their most specific feature, i.e. that of driving the statevectorof any individual physical system into appropriate and physicallymeaningful manifolds, they allow for a different interpretation. Onecould even say (if one wants to avoid that they too, as the standardtheory, speak only of what we find) that theyrequire a different interpretation, one that accounts for ourperceptions at the appropriate, i.e. macroscopic, level.We must admit that this opinion is not universally shared. Accordingto various authors, the ‘rules of the game’ embodied inthe precise formulation of the GRW and CSL theories represent allthere is to say about them. However, this cannot be the whole story:stricter and more precise requirements than the purely formal onesmust be imposed for a theory to be taken seriously as a fundamentaldescription of natural processes (an opinion shared by J. Bell). Thisrequest of going beyond the purely formal aspects of a theoreticalscheme has been denoted as (the necessity of specifying) the PrimitiveOntology (PO) of the theory in an extremely interesting recent paper[Allori, 2007]. The fundamental requisite of the PO is that it shouldmake absolutely precise what the theory is fundamentally about.This is not a new problem; as already mentioned it has been raised byJ. Bell since his first presentation of the GRW theory. Let mesummarize the terms of the debate. Given that the wavefunction of amany-particle system lives in a (high-dimensional) configurationspace, which is not endowed with a direct physical meaning connectedto our experience of the world around us, Bell wanted to identify the‘local beables’ of the theory, the quantities on which onecould base a description of the perceived reality in ordinarythree-dimensional space. In the specific context of QMSL, he [Bell1987, p. 45] suggested that the ‘GRW jumps’, which wecalled ‘hittings’, could play this role. In fact theyoccur at precise times in precise positions of the three-dimensionalspace. As suggested in [Allori, 2007] we will denote this positionconcerning the PO of the GRW theory as the ‘flashesontology.’However, later, Bell himself suggested that the most naturalinterpretation of the wavefunction in the context of a collapse theorywould be that it describes the ‘density […] ofstuff’ in the 3N-dimensional configuration space [Bell, 1990,p. 30], the natural mathematical framework for describing a system ofN particles. The authors of ref. [Allori, 2007] appropriately havepointed out that this position amounts to avoid to committing oneselfabout the PO ontology of the theory and, consequently, to leave vaguethe precise and meaningful connections it allows to establish betweenthe mathematical description of the unfolding of physical processesand our perception of them.The interpretation which, in the opinion of the present writer, ismost appropriate for collapse theories, has been proposed in series ofpapers [Ghirardi, Grassi and Benatti, 1995, Ghirardi, 1997a, 1997b]and has been referred in [Allori, 2007] as ‘the mass densityontology’. Let us briefly describe it.First of all, various investigations [Pearle and Squires, 1994] hadmade clear that QMSL and CSL needed a modification, i.e., thecharacteristic localization frequency of the elementary constituents ofmatter had to be made proportional to the mass characterizing theparticle under consideration. In particular, the original frequency forthe hitting processes f = 10-16 sec-1 isthe one characterizing the nucleons, while, e.g., electrons wouldsuffer hittings with a frequency reduced by about 2000 times.Unfortunately we have no space to discuss here the physical reasonswhich make this choice appropriate; we refer the reader to the abovepaper, as well as to the recent detailed analysis by Peruzzi and Rimini[Peruzzi and Rimini, 2000]. With this modification, what the nonlineardynamics strives to make ‘objectively definite’ is the mass distribution in the whole universe. Second, a deep criticalreconsideration [Ghirardi, Grassi and Benatti, 1995] has made evidenthow the concept of ‘distance’ that characterizes theHilbert space is inappropriate in accounting for the similarity ordifference between macroscopic situations. Just to give a convincingexample, consider three states |h>, |h*> and|t> of a macrosystem (let us say a massive macroscopicbulk of matter), the first corresponding to its being located here,the second to its having the same location but one of its atoms (ormolecules) being in a state orthogonal to the corresponding state in|h>, and the third having exactly the same internal stateof the first but being differently located (there). Then, despite thefact that the first two states are indistinguishable from each otherat the macrolevel, while the first and the third correspond tocompletely different and directly perceivable situations, the Hilbertspace distance between |h> and |h*>, is equalto that between |h> and |t>.When the localization frequency is related to the mass of theconstituents, then, in completely generality (i.e. even when one isdealing with a body which is not almost rigid, such as a gas or acloud), the mechanism leading to the suppression of the superpositionsof macroscopically different states is fundamentally governed by thethe integral of the squared differences of the mass densitiesassociated to the two superposed states. Actually, in the originalpaper [Ghirardi, Grassi and Benatti, 1995] the mass density at a pointwas identified with its average over the characteristic volume of thetheory, i.e., 10-15 cm 3 around that point. Itis however easy to convince onself that there is no need to do so[Ghirardi, 2007] and that the mass density at any point, directlyidentified by the statevector (see below), is the appropriate quantityon which to base an appropriate ontology. Accordingly, we take thefollowing attitude: what the theory is about, what is real ‘outthere’ at a given space point x, isjust a field, i.e. a variable m(x,t) givenby the expectation value of the mass density operatorM(x) at xobtained by multiplying the mass of any kind of particle times thenumber density operator for the considered type of particle andsumming over all possible types of particles which can be present:(7)m(x,t) =< F,t |M(x) |F,t >; M(x)=Sum(k)m(k)a*(k)(x)a(k)(x).Here |F,t> is the statevector characterizing thesystem at the given time, anda*(k)(x) anda(k)(x) are the creationand annhilation operators for a particle of type k at pointx. It is obvious that within standardquantum mechanics such a function cannot be endowed with any objectivephysical meaning due to the occurrence of linear superpositions whichgive rise to values that do not correspond to what we find in ameasurement process or what we perceive. In the case of GRW or CSLtheories, if one considers only the states allowed by the dynamics onecan give a description of the world in terms ofm(x,t), i.e., one recoversa physically meaningful account of physical reality in the usual3-dimensional space and time. To illustrate this crucial point weconsider, first of all, the embarrassing situation of a macroscopicobject in the superposition of two differently located positionstates. We have then simply to recall that in a collapse modelrelating reductions to mass density differences, the dynamicssuppresses in extremely short times the embarrassing superpositions ofsuch states to recover the mass distribution corresponding to ourperceptions. Let us come now to a microsystem and let us consider theequal weight superposition of two states |h> and|t> describing a microscopic particle in two differentlocations. Such a state gives rise to a mass distributioncorresponding to 1/2 of the mass of the particle in the two consideredspace regions. This seems, at first sight, to contradict what isrevealed by any measurement process. But in such a case we know thatthe theory implies that the dynamics running all natural processeswithin GRW ensures that whenever one tries to locate the particle hewill always find it in a definite position, i.e., one and only one ofthe Geiger counters which might be triggered by the passage of theproton will fire, just because a superposition of ‘a counterwhich has fired’ and ‘one which has not fired ’ isdynamically forbidden.This analysis shows that one can consider at all levels (the micro andthe macroscopic ones) the fieldm(x,t) as accounting for‘what is out there’, as originally suggested bySchrödinger with his realistic interpretation of the square ofthe wave function of a particle as representing the ‘ fuzzy’ character of the mass (or charge) of the particle. Obviously,within standard quantum mechanics such a position cannot be mantainedbecause ‘wavepackets diffuse, and with the passage of timebecome infinitely extended … but however far the wavefunctionhas extended, the reaction of a detector … remainsspotty’, as appropriately remarked in [Bell, 1990]. As we hopeto have made clear, the picture is radically different when one takesinto account the new dynamics which succeeds perfectly in reconcilingthe spread and sharp features of the wavefunction and of the detectionprocess, respectively.It is also extremely important to stress that, by resorting to thequantity (7) one can define an appropriate ‘distance’between two states as the integral over the whole 3-dimensional spaceof the square of the difference ofm(x,t) for the two givenstates, a quantity which turns out to be perfectly appropriate toground the concept of macroscopically similar or distinguishableHilbert space states. In turn, this distance can be used as a basis todefine a sensible psychophysical correspondence within the theory.12. The Problem of the Tails of the Wave FunctionIn recent years, there has been a lively debate around a problem whichhas its origin, according to some of the authors which have raised it,in the fact that even though the localization process whichcorresponds to multiplying the wave function times a Gaussian and thuslead to wave functions strongly peaked around the position of thehitting, they allow nevertheless the final wavefuntion to be differentfrom zero over the whole of space. The first criticism of this kindwas raised by A. Shimony [Shimony, 1990] and can besummarized by his sentence,one should not tolerate tails in wave functions which areso broad that their different parts can be discriminated by the senses,even if very low probability amplitude is assigned tothem.After a localization of a macroscopic system, typically the pointer ofthe apparatus, its centre of mass will be associated to a wavefunction which is different from zero over the whole space. If oneadopts the probabilistic interpretation of the standard theory, thismeans that even when the measurement process is over, there is anonzero (even though extremely small) probability of finding itspointer in an arbitrary position, instead of the one corresponding tothe registered outcome. This is taken as inacceptable, as indicatingthat the DRP does not actually overcome the macro-objectificationproblem.Let us state immediately that the (alleged) problem arises entirelyfrom keeping the standard interpretation of the wave functionunchanged, in particular assuming that its modulus squared gives theprobability density of the position variable. However, as we havediscussed in the previous section, there are much more serious reasonsof principle which require to abandon the probabilistic interpretationand replace it either with the ‘flash ontology’, or withthe ‘ mass density ontology’ which we have discussedabove.Before entering into a detailed discussion of this subtle point weneed to focus better the problem. We cannot avoid making two remarks.Suppose one adopts, for the moment, the conventional quantum position.We agree that, within such a framework, the fact that wave functionsnever have strictly compact spatial support can be consideredpuzzling. However this is an unavoidable problem arising directlyfrom the mathematical features (spreading of wave functions) and fromthe probabilistic interpretation of the theory, and not at all aproblem peculiar to the dynamical reduction models. Indeed, the factthat, e.g., the wave function of the centre of mass of a pointer or ofa table has not a compact support has never been taken to be a problemfor standard quantum mechanics. When, e.g., the wave function of atable is extremely well peaked around a given point in space, it hasalways been accepted that it describes a table located at a certainposition, and that this corresponds in some way to our perception ofit. It is obviously true that, for the given wave function, thequantum rules entail that if a measurement were performed the tablecould be found (with an extremely small probability) to be kilometersfar away, but this is not the measurement or themacro-objectification problem of the standard theory. The latterconcerns a completely different situation, i.e., that in which one isconfronted with a superposition with comparable weights of twomacroscopically separated wave functions, both of which possess tails(i.e., have non-compact support) but are appreciably different fromzero only in far-away narrow intervals. This is the reallyembarrassing situation which conventional quantum mechanics is unableto make understandable. To which perception of the position of thepointer (of the table) does this wave function correspond?The implications for this problem of the adoption of the QMSL theoryshould be obvious. Within GRW, the superposition of two states which,when considered individually, are assumed to lead to different anddefinite perceptions of macroscopic locations, are dynamicallyforbidden. If some process tends to produce such superpositions, thenthe reducing dynamics induces the localization of the centre of mass(the associated wave function being appreciably different from zeroonly in a narrow and precise interval). Correspondingly, thepossibility arises of attributing to the system the property of beingin a definite place and thus of accounting for our definite perceptionof it. Summarizing, we stress once more that the criticism about thetails as well as the requirement that the appearance ofmacroscopically extended (even though extremely small) tails bestrictly forbidden is exclusively motivated by uncritically committingoneself to the probabilistic interpretation of the theory, even forwhat concerns the psycho-physical correspondence: when this positionis taken, states assigning non-exactly vanishing probabilities todifferent outcomes of position measurements should correspond toambiguous perceptions about these positions. Since neither within thestandard formalism nor within the framework of dynamical reductionmodels a wave function can have compact support, taking such aposition leads to conclude that it is just the Hilbert spacedescription of physical systems which has to be given up.It ought to be stressed that there is nothing in the GRW theorywhich would make the choice of functions with compact supportproblematic for the purpose of the localizations, but it also has to benoted that following this line would be totally useless: since theevolution equation contains the kinetic energy term, any function, evenif it has compact support at a given time, will instantaneously spreadacquiring a tail extending over the whole of space. If one sticks tothe probabilistic intepretation and one accepts the completeness of thedescription of the states of physical systems in terms of the wavefunction, the tail problem cannot be avoided.The solution to the tails problem can only derive from abandoningcompletely the probabilistic interpretation and from adopting a morephysical and realistic intepretation relating ‘what is outthere’ to, e.g., the mass density distribution over the wholeuniverse. In this connection, the following example will be instructive[Ghirardi, Grassi and Benatti, 1995]. Take a massive sphere of normaldensity and mass of about 1 kg. Classically, the mass of this bodywould be totally concentrated within the radius of the sphere, call itr. In QMSL, after the extremely short time interval in whichthe collapse dynamics leads to a ‘regime’ situation, and ifone considers a sphere with radius r + 10-5 cm, theintegral of the mass density over the rest of space turns out to be anincredibly small fraction (of the order of 1 over 10 to the power1015) of the mass of a single proton. In such conditions, itseems quite legitimate to claim that the macroscopic body is localisedwithin the sphere.However, also this quite reasonable position has been questioned andit has been claimed [Lewis, 1997], that the very existence of the tailsimplies that the enumeration principle (i.e. the fact that the claim‘particle 1 is within this box & particle 2 is within thisbox & … & particle n is within this box& no other particle is within this box’ implies the claim‘there are n particles within this box’) does nothold, if one takes seriously the mass density interpretation ofcollapse theories. This paper has given rise to a long debate whichwould be inappropriate to reproduce here. We refer the reader to thefollowing papers [Ghirardi and Bassi, 1999; Clifton and Monton, 1999a,1999b; Bassi and Ghirardi, 1999, 2001]. Various arguments have beenpresented in favour and against the criticism by Lewis.We would like to conclude this brief analysis by stressing once morethat, in our opinion, all the disagreements and the misunderstandingsconcerning this problem have their origin in the fact that the ideathat the probabilistic interpretation of the wave function must beabandoned has not been fully accepted by the authors who find somedifficulties in the proposed mass density intepretation of theCollapse Theories. For a recent reconsideration of the problem werefer the reader to the paper by [Lewis, 2003]. 13. The Status of Collapse Models and Recent Positions about themWe recall that, as stated in Section 3, the macro-objectificationproblem has been at the centre of the most lively and most challengingdebate originated by the quantum view of natural processes. Accordingto the majority of those who adhere to the orthodox position such aproblem does not deserve a particular attention: classical conceptsare a logical prerequisite for the very formulation of quantummechanics and, consequently, the measurement process itself, thedividing line between the quantum and the classical world, cannot andmust not be investigated, but simply accepted. This position has beenlucidly sunmmarized by J. Bell himself [Bell, 1981]: Making a virtue of necessity and influenced bypositivistic and instrumentalist philosophies, many came to hold notonly that it is difficult to find a coherent picture but that it iswrong to look for one - if not actually immoral then certainlyunprofessionalThe situation has seen many changes in the course of time, and thenecessity of making a clear distinction between what is quantum andwhat is classical has given rise to many proposals for ‘easysolutions’ to the problem which are based on the possibility,for all practical purposes (FAPP), to locate the splittingbetween these two faces of reality at different levels.Then came Bohmian mechanics, a theory which has made clear, in a lucidand perfectly consistent way, that there is no reason of principlerequiring a dichotomic description of the world. A universal dynamicalprinciple runs all physical processes and even though ‘itcompletely agrees with standard quantum predictions ’ it implieswave-packet reduction in micro-macro interactions and the classicalbehaviour of classical objects.As we have mentioned, the other consistent proposal, at thenonrelativistic level, of a conceptually satisfactory solution of themacro-objectification problem is represented by the Collapse Theorieswhich are the subject of these pages. Contrary to bohmian mechanics,they are rival theory of quantum mechanics, since they make differentpredictions (even though quite difficult to put into evidence)concerning various physical processes.Let us now analyze some of the recent critical positions concerningthe two just mentioned approaches (in what follows I will takeadvantage of the nice analysis of a paper which I have been asked toreferee and of which I do not know the author). Various physicistshave criticized Bohm approach on the basis that, being empiricallyindistinguishable from quantum mechanics, such an approach is anexample of ‘bad science ’ or of ‘a degenerateresearch program ’. Useless to say, I do not consider suchcriticisms as appropriate; the conceptual advantages and the internalconsistency of the approach render it an extremely appealingtheoretical scheme (incidentally, one should not forget that it hasbeen just the critical investigations on such a theory which have ledBell to derive his famous and conceptually extremely relevantinequality).This being the situation, one would think that theories like the GRWmodel would be exempt from an analogous charge, since they actuallyare (in principle) empirically different from the standard theory. Forinstance they disagree from such a theory since they forbid theoccurrence of macroscopic massive entangled states. In spite of this,they have been the object of an analogous attach by the adherents tothe ‘new orthodoxy ’ [Bub 1997, Joos et al.,1996, Zurek, 1993] pointing out that environmental induced decoherenceshows that, FAPP, collapse theories are simply phenomenologicalaccounts of the reduced state to which one has to resort since one hasno control of the degrees of freedom of the environment. When onetakes such a position, one is claiming that, essentially, GRW cannotbe taken as a fundamental description of nature, mainly because itsuffers from the limitation of being empirically indistinguishablefrom the standard theory, provided such a theory is correctly appliedtaking into account the actual physical situation. Also in this case,and even at the level at which such an analysis is performed, I do notthink that the practical indistinguishability from the standardapproach is a sufficient reason not to take seriously collapsemodels. In fact, there are many very well known and compelling reasons(see, e.g. [Bassi, 2000 and Adler, 2003] to prefer a logicallyconsistent unified theory to one which makes sense only due to thealleged practical impossibility of detecting thesuperpositions of macroscopically distinguishable states.But this is not the whole story. Another criticism, aimed to‘deny’ the potential interest of collapse theories makesreference to the fact that within any such theory the ensuing dynamicsfor the statistical operator can be considered as the reduced dynamicsderiving from a unitary (and, consequently, essentially a standardquantum) dynamics for the states of an enlarged Hilbert space of acomposite quantum system S+E involving, besides the physicalsystem S of interest, an ancilla E whose degrees offreedom are completely unaccessible:due to the quantum dynamicalsemigroup nature of the evolution equation for the statisticaloperator, any GRW-like model can always be seen as a phenomenologicalmodel deriving from a standard quantum evolution on a larger Hilbertspace. In this way, the unitary deterministic evolution characterizingquantum mechanics would be fully restored.Apart from the obvious remark that such a critical attitude completelyfails in grasping and purposedly ignores the most important feature ofcollapse theories, i.e. of dealing with individual quantum systems andnot with statistical ensembles and of yielding a perfectlysatisfactory description, matching our perceptions concerningindividual macroscopic systems, I believe that invoking anunaccessible ancilla to account for the nonlinear and stochasticcharacter of GRW-type theories, is once more a purely verbal way ofavoiding to face the real puzzling aspects of the quantum descriptionof macroscopic systems. And this is not the only negative aspect ofsuch a position; any attempt considering legitimate to introduceunaccessible entities in the theory, when one takes into considerationthat there are infinitely possible and inequivalent ways of doingthis, amounts really to embarking oneself in a ‘degenerateresearch program ’.Other reasons for ignoring the dynamical reduction program have beenput forward recently by the community of scientists involved in theinteresting and exciting field of quantum information. I will notspend too much time in analyzing and discussing the new position aboutthe foundational issues which have motivated the elaboration ofcollapse theories. The crucial fact is that, from this perspective,one takes the theory not to be about something real ‘occuringout there ’ in a real word, but simply about information. Thispoint is made extremely explicit in a recent paper [Zeilinger,2006]: information is the most basic notion of quantummechanics, and it is information about possible measurement resultsthat is represented in the quantum state. Measurement results arenothing more than state of the classical apparatus used by theexperimentalist. The quantum system then is nothing other than theconsistently constructed referent of the information represented inthe quantum state.It is clear that if one takes such a position almost all motivationsto be worried by the measurement problem disappear, and with them thereasons to work out what Bell has denoted as ‘an exact versionof quantum mechanics ’. Accordingly I believe that the mostappropriate reply to this type of criticisms is to recall that J. Bell[Bell, 1990] has included ‘information’ among the wordswhich must have no place in a formulation with any pretension tophysical precision. In particular he has stressed that one cannot evenmention information unless one has given a precise answer to the twofollowing questions: Whose information? and Informationabout what?A much more serious attitude, in our opinion, is to call attention, asmany serious authors do, on the fact that since collapse theoriesrepresent rival theories with respect to standard quantum mechanicsthey lead to the identification of experimental situations which wouldallow, in principle, crucial tests to discriminate between the two. Aswe have discussed above, presently such tests seems not easilyfeasible, but an analysis of the suggested tests we have mentioned,shows that such tests are not completely out of reach as soon as sometecnological improvements in dealing with mesoscopic systems willbecome available.SummaryWe hope to have succeeded in giving a clear picture of the ideas, theimplications, the achievements and the problems of the DRP. 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